The radial engine idea is very simple -- it takes the pistons and arranges them in a circle around the crankshaft, as shown here:
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Image courtesy Baris Mengutay
You can see in the illustration that this is a five-cylinder engine -- radial engines typically have anywhere from three to nine cylinders. The radial engine has the same sort of pistons, valves and spark plugs that any four-stroke engine has. The big difference is in the crankshaft.
Instead of the long shaft that's used in a multi-cylinder car engine, there is a single hub -- all of the piston's connecting rods connect to this hub. One rod is fixed, and it is generally known as the master rod. The others are called articulating rods. They mount on pins that allow them to rotate as the crankshaft and the pistons move.